Three-dimensional mathematical demonstrator



J2 17m g 7 S hetsShee t 2 F. AR'MSTEAD THREE-DIMENSIONAL MATHEMATICAL DEMONSTRATOR Nov. 6, 1951 Filed u 7, 1947 (Ittomeg Q 6, 1951 F. PfARMSTEAD 2,573,946 I THREE-DIMENSIONAL MATHEMATICAL DEMONSTRATOR Fi led Aug. '7, 1947 v 7 Sheets-Shet s,

. INVENTORQ 41710 3. mmsrmfl 1951 F P. ARMSTEAD A THREE-DIMENSIONAL MATHEMATICAL DEMONSTRATOR Filed Aug. 7, 1947 7 Sheets-Sheet 4 INVENTOR. 1 14am EHEMETL-ZHU Nov. 6, 1951 F. P. ARMSTEAD 2,573,946

THREE-DIMENSIONAL MATHEMATICAL DEMONSTRATOR Filed Aug. '7, 1947 '7 Sheets-Sheet 5 axe 56 54 INVENTOR.

Nov. 6, 1951 F. P. ARMSTEAD THREE-DIMENSIONAL MATHEMATICAL DEMONSTRATOR '7 Sheets-Sheet 6 Filed Aug. 7, 1947 (Ittomeg Nov. 6, 1951 F. P. ARM-STEAD 2,573,946

THREE-DIMENSIONAL MATHEMATICAL DEMONSTRATOR Filed Aug. 7, 1947 7 Shets-Sheet 7 Gttorneg Patented Nov. 6, 15251 UNI-TED- STATESPATENT OFFICE Tnn npnu nsroNAL MATHEMATICAL DEMQNSTRATOR.

30m ns 1,.

The present invention relates, to, a three. mensional mathematical demonstrator.

The principal purpose t my invention is to provide means tier demonstrating to students the basic concepts of geometric. figures- The traditional methocL of illustrating geometric, figures on a blackboard is somewhat, inadequate because the fore-shortened lines resulting from repree senting althree dimensional; figure on a two, dimensional planeconiussmany students. There.

die

is alwaysacertain amountvof optical;illusionv in viewing these single plane. representations of solids.

More particularly, it isagpur-pose of my invention to'provide a demonstrator thatvwouldshow in three dimensions. therlelative position, of lines, planes, polyhedra, cylinders, and eoneslin. space. This. mechanical. device as contemplated illus-.

trates the figures necessary for the solutionof most of the basic theoriesland corollaries used. in

the study ofsolid geometry.

It is also a purpose of my invention to p vide a demonstrator of the character above re,-

ferredto. which will have a range offfigur'es, and.

spatial. relationships adequateto present-most of the types of, problems in solidlgeometry,v excepting those relatingtoasphere The-demonstrator isconstructed'to allow. for easy trans-ition irom the illustrationiof one. shapeor figureto. the illus.-.j tration. ofanother figure of theisametype. The-- figuresas demonstrated are transparent or I open to allow for. internal viewing so that the. intersections. of lines and planes-,andtherelative positionsof inscribedfigures may be, Seen.

A, furtherv purposeot'my invention is, to pro vide in a demonstrator ofthe. character described, a suitable base. having a plane. surface to permit the marking. of figuresor'partsoi'fig ures thereon and suitable labelingthereof; in-

combination with means to support, above. the base, amultiplicity of flexible linesin close'rela' tion and additional plane surface" elements to demonstrate" the appearance of the figures under discussion. The entire device, incl'udingthe movable supporting parts and" the flexible construction members, is of such construction as to per.- mit' a quick change from-'onetype of" figure to another type of figure so that" the amount of timenecessary to'set" up a required solid; on a demonstrator is generallyiess than would be" re-' quired-to draw th'e' three dimensional solid on the blackboard:

invention to providea mathematical demonstrae; tor consisting of a table section divided into two, removable section's either of which maybe used as a surface to write upon with chalk, the remov-- able sections containing holes-arranged in definite geometrical patterns through which are led construction strings to outline the geometric ures, in combination with'a support for the table including. a rack of tubes each-of theremovable sections for containingandguiding the construction strings-andsmallv weights. to hold them; place; Overhead holding means'are secured to: the table by quickly removable supportingrods/ to. which the holding means are attached;

Thev novel featuresthatl consider characteris,-.- tic: ofvmy, invention-are set forth with partia larity in. the, claims The invention itself, however, both asto its-organization and its method oi operation, together with additional objects and, advantages thereof, will be understoodironr-the description of av specific embodiment when-read in connection with theaccompanying drawings. in'which: a V Figure 1 isa. plan view of the device-embodying my. invention; 7 Figure. 2Iis aplanview with the top portion" of the device removed to illustrate theconstruc-w tionibeneath the. top portion;

Figure, 3'. avertical'section taken on the line 3-3of, Fi urez; V v

Figure 4'is ai -fragmentary sectional View of the stringmounting frame of-the device;

Figure5 is an enlargedffragmentary view-illus-. trating. how the strings are held down in place; Figure 6 is a perspective view illustrating the formatiqn of a. geometrical figure by means or. the. strings and the, supporting rod structure on the-device;

Figure '7 is a fragmentary view of the-supports ing rod mechanism illustrating the manner. in}

discs,; squares,- eto., tourecei ve the strings; and".

for attachment to the supporting rods;

Figure 9-is a view like, Figure fi'showing a dif ferent type of solid demonstrated;

Figure" 10 is a plan viewof a modified-"tube:

. rackto demonstrateel lilcitica'l bodies;

It isafurther and more specific object-of" niv" Fig ureii is'a view, in side elevation; of a desk' type of; unit and Figure 12 isfa detail view or the stringholding;

dBviceMsedwithFigure 10.

Referring now to the drawings, and particularly to Figures 1, 2 and 3, the basic construction of the device embodies a portable rectangular housing III, the base of which may be provided with suit able casters as shown so that it may be moved around on the floor. The housing has an open to which is adapted to receive two removable sections I I and I2 which are provided with a coating such as is used for blackboards. Each of the sections II and I2 is preferably made of a suitable sheet material such as plywood or a composition sheet and a supporting frame. The section II is shown as provided with a number of drilled holes I3, most of which are arranged in pairs but some of which are single holes spaced substantially from the adjacent holes. The holes are arranged in definite geometric patterns through which are led strings I4 (see Figure 5), the strings being used to outline the figures in designs .to be demonstrated. The ends of the stringsare knotted or provided with any suitable head to keep them from falling through the holes.

Within the housing, directly beneath each unit of the top, is a series of tube guides. comprise three spaced parallel panels IE, IT and I8 provided with suitable holes I9 therethrough to receive tubes 2|. Some of the holes in the guides are large enough to receive two tubes 2! side by side while other holes are large enough only for a single tube as shown at 20 in Figure 3 of the drawings. The tubes 2! are all alike and extend through the holes in the guides to rest upon the fioor of the housing It. Within each tube there is a small weight 22 to which a string I4 is attached and the string is led upwardly from the tube through the corresponding opening I3 and the top II where it is knotted or headed so as to prevent its passing down through the opening I3 all of the way. The table top section II or I2 can be lifted with a set of strings and removed in its entirety, leaving the tubes 2| in the guides. The three panels I6, I1 and I8 are fastened together by suitable bolts 24 and spacer 25. In order to mount strings above the table tops, a supporting frame I la is provided for the removable table tops H and I2. The frame isremovably mounted on the housing III by a peripheral rib Illa on the top of the housing I!) and'a corresponding groove in the frame Ila. Ateach end of the'housing I0 mounting blocks 26 and 21 are provided. These blocks are provided with sockets 28 and 29 adapted to receive upright posts 30 and 3I, the lower ends of which are reduced in diameter so as to engage in the sockets. The posts 30 and 3 I receive Starrett sleeve clamps 32 and 33 which in turn are adapted to clamp supporting bars such as the main cross bar 34 or a hook bar 35 at the desired position on the posts. The suspension bars 36 and 31 are provided with suitable clamps 38 and 39 for clamping on the horizontal bar 34. The suspension bars 36 and 31 may take any suitable shape or design. For example, in Figure 6 I have shown the suspension bar as straight with a fixed collar 40 thereon, a threaded portion 4| and a hook 42. The threaded portion is adapted to receive a wing nut 43, similar to the nut shown in Figure '7. The hook bar 35, likewise, is provided with a fixed collar 44, a threaded portion 45 and a hook 46 to engage the strings. The hooks preferably are made as shown in Figure 7 so that a number of strings can be engaged with the knots holding the strings against removal.

In order to set up a figure, for example, a pyramid as shown in Figure 6, four strings are drawn upwardly from certain of the apertures 13 and The guides 4 hooked into the hook 42 which is suspended by the bar 36. In the illustration of Figure 6, a central string is also brought up to the hook 42 and secured in place. In Figure 6 the demonstration is of a plane intersecting a pyramid parallel to the base. The plane may be any suitable sheet of material 41 with slots in the edges and. corners thereof as indicated at 48 (see Figure 8). The sheet 4! is also apertured at various places to permit mounting it on one of the bars 35 by passing the threaded portion of thebar through an opening in the sheet and clamping the sheet against the collar 44 by means of the wing nut 43.

.It will be understood, of course, that various shapes of the sheets are provided, such for example, as squares, rectangles, ellipses and the like for demonstrating the intersection of various solid figures by a plane. The sheets 41, like the table tops [I and I2, are preferably coated suitably so that they may be used to mark line with 3 chalk thereon like the usual blackboard. It is believed to be evident that variou relationships of the parts of the quadrangular right pyramid shown in Figure 6 can be demonstrated very effectively with the device. It is easy to show the relation of a line A--O to another line AO'.

In Figure 9 I'have shown another demonstration in which the figure is an oblique prism. The sheet 41 in this instance is used for the top of the figure. The lower end of the prism is outlined in chalk on the table top II. The strings IQ for the four longitudinal edges of the prism are drawn up to and secured on the sheet 41. Also the four diagonals of the prism are formed by using the other strings at each of the points A, B, C, D. By this demonstration the student can easily see that the diagonals intersect at a point which is the center of the prism. The diagonal strings I4 form two like oblique pyramids within the prism. It is easy, with this device, to demonstrate the various propositions relating to the prism shown.

In Figure 10 of the drawings the top of a tube rack I5 with a modified pattern is shown. In this rack the tubes 2 I arearrang ed for the demonstration of elliptical bodies. The rack is out out on one side at 50 so it may be mounted on either side of the housing. A suitable top section having corresponding string openings is of course provided for cooperation with the rack I5.

Figures 11 and 12 illustrate a modification which is particularly adapted for use on a table. In this form of the invention a shallow housing 5| is provided with a] top plate 52 which has string apertures 53 therein for strings 54. In this form of the invention the strings are wound on small spools 55 which are mounted on the bottom wall 56 of the housing 5|. The spool (see Figure 12) is journalledon a shaft 51 which is fastened in standards 58 and 59. A coiled spring fiilis mounted inside the spool with one end 61 fastened to the shaft 5] and the other end 62 fastened to the spool. The spring serves to keep the strings drawn down into the housing but permits them to be pulled out for making the figures for demonstration. The use of the table model is the same as the main form of the invention.

The holes I3 provided in the table tops I I and I2 are arranged in the proper geometric patterns to permit a demonstration of a great variety of non-spherical solid figures such as are conventhe following list of various. constructions of pyramids, cylinders 'and cones isgiven as-amartia-l roup of thefigures that may be demonstrated.

} 1; Prisms Right prisms such as triangular, quadranguIa'r, hexagonal and octagonal right prisms 1 B. Oblique prisms such as triangular, quadrangular (parallelepiped) hexagonal and octagonal, oblique prisms C. "Truncated prisms, either right or oblique "2.='Pyramids; triangular, quadrangular, hexagonal and octagonal A. Right pyramids Bl Oblique pyramids C. Frustums'of right pyramids D. Frustums of oblique pyramids "3. Cylinders; right and oblique fAI Circular cylinders -1 T: B; Elliptical cylinders -4. Cones;"right and oblique, (circular and --;elliptical) a --'5. 'Conic sections, parabolas, hyperbolas, el-

" 'lipses, and circles -6. Inscribed polyhedra such as:

A. Prisms inscribed in cylinders B. Pyramids inscribed in cylinders C. Pyramids inscribed in cones D. Frustums of pyramids inscribed in frustums of cones. '7. Circumscribed-polyhedra such as: l -A-. Prisms circumscribed about cylinders B.Prisms circumscribed about cones C. Pyramids circumscribed about cones D. Frustums of pyramids circumscribed about frustums of cones.

'8; Pyramids inscribed in prisms 9, Cones inscribed in cylinders.

Theabove list is, of course, only partial. More specific forms of the above general classes of figures with variations in angles and dimensions may be demonstrated easily. Many theorems and corollaries may be shownwith the demonstrator. Those. theorems and corollaries, concerning-lines and planes and the various solids hereinbefore mentioned may be shown.-

Experience has-shown that the three-dimension-al mathaticaldemonstrator can be used in theclassroom to make more meaningful the basic concepts-employedin solid geometry. Either student or teacher can construct the-desired figures ordemonstrate relationships. Students appear gradually to overcome their inability toyisua-lizeand; rep-resent the relative positions on the demonstrator-of objects inspace. Asa result, they develop a knowledge of familiar spatial. relationships. Once the student has this knowledge; or ifj'the teacher wants to test for understanding, the student can build the figures himself.

The optical illustionexperienced when viewing single plane representations of-solidsadisap pears almost completely whenzcach figure is set up and portrayed in its. true shape. By the use of the demonstrator, it is possible to construct or to'have constructed the figures as they nat urally appear; and it-doesnot-require the utilization of the somewhat questionable skill of an average instructor toportray the figureson the blackboard. As pointed out earlier, blackboard drawings are fiat, therefore, it naturally follows that projections are necessary in order to show all the sides of the figures. On the other hand, in drawing projections on the blackboard it is difficult for the student to realize exactly the true shape of the figures unless the instructor is particula'rly skilled in both (a) drawing and (b) sex-- planations. In vthe usual course or events, the drawings 'must.-;be put :on :hur'rie'dly to conserve time, and may be inaccurate, with the result that the :drawings; may confuse .rather than clarify. The economy of time spent in construction-and the effective accuracy of ,"the demonstrator rovercomes thesepdifliculties.

Theoutstanding advantages of the demonstra tor-are'its ,fiexibility, 'wide variety of figures and relationships, its transparency, simplicity, and suitability to student use.

.jBy the use of the demonstrator it is made possible'to see all sides of the figure without distortion, and in addition thereto, it is made possible to show imaginary lines and planes as represented by the 'yariousstrings and flat inserts which may be -set up as necessary. In short, solids are solid.

They appear-in but one shape, the shape in which theyrare made. But in geometrythere is need for cross-sections, the superimposition of figures, the inscribing and circumscribing of figures, the gradual change from one type or shape of figure to-another. These are limitations for the solid models and advantages for the demonstrator. Blackboard drawings and conventional textbook drawings can illustrate the various cross-sections, lines and relationships, with the use of man-y foreshortened and dot-ted lines For somestudents'these drawings may heighten the confusion of concepts in the early stages of learning. .Suchmisconceptions may result in faulty learning or in .poor' attitudes. Although threedimensional pictures as represented by the stereograph .do not rely upon the devices of foreshortened lines, etc., to show three dimensions, they have in common with other drawings and illust'ra'tions the limitation of inflexibility. The demonstrator is versatile enough not only to allow the showing of guide and imaginary lines as mentioned, but' also to show cross-sections of figures inalmost unlimited numbers. The type of'cross-section used is entirely within the control oflthe instructor operating the demonstrator. Since cross-sections are imperative to proper understanding of solid geometry, it follows that the demonstrator becomes? a valuable teaching aid in arriving at that objective. One of the strongest appeals of the demonstra tor is' its use by students to construct and clarify their concepts, Not infrequently, a student will challenge the accuracy of a statement of a theorem or figure, which question can usually be answered by means of measurement or construction. It is desirable to establish a certain amount,

tionally upset. As mentioned above, blackboard.

drawings do not lend themselves to measurement,

-since' aswe have seen, solids can only be drawn thereon; by projection methods. On the other hand, if solids are used, measurements are'limited to outside-surfaces, and certain internal measurements which can be calculated only by the use of certain formulas. But with the demonstrator it is possible to make internal measurements of lines, cross-sections, guide lines, superimposed, circumscribed and inscribed figures. The advantage should be easily recognized.

The wide range of figures and spatial relationships that can be handled by the demonstrator makes it possible to present practically any type of problem met in the study of lines, planes, polyhedra, inscribed and circumscribed polyhedra, circular cylinders and cones, and elliptical cylinders and cones. I

The demonstrator is designed to illustrate two separate figures at the same time. The arrangement of the strings on both units are'such that two figures that are congruent or similar to each other can be set up. For example, it would be possible to set up two hexagonal pyramids that are congruent or similar to one another, or it would be possible to illustrate a right hexagonal prism on one unit and an oblique circular cylinder on the other.

A unique feature of the flexibility of the demonstrator is the speed and ease of transition from one figure to another. For example, a right quadrangular prism can be transformed into an oblique quadrangular prism by loosening the Starrett sleeve and sliding it along the horizontal bar. The student can watch the figure change shape in much the same manner as he sees movement in a motion picture. It can be illustrated that the angle made by the lateral edges (strings) with the lower base is determined by the distance the sleeve is moved along the horizontal rod. If the upper base (disc) is held at the same distance and parallel from the lower base, the volume of the figure does not change even though the figure has been changed from a,

right to an oblique prism. This is true, for the volume of a prism is determined by the area of the base times the altitude, and these two factors can be held constant. .If the rod support-.

ing the disc is moved in a vertical direction, either up or down, the volume of the'figure can be increased or decreased as the, altitude is changed. It is possible to move in both a horizontal and a vertical direction or at an angle to either of them. For example, by tilting the upper base of a right prism in such a way that it is not parallel to the lower base, one will have a truncated prism. a

To create new figures, the discs can be. rotated in a clockwise or counter-clockwise direction. These figures can be changed as to volume and configuration by lowering or raising the disc or increasing the turning of the disc.

Having thus described my invention, I claim:

1. A device for demonstrating geometric figures comprising in combination, a housing, a frame work therein comprising a plurality of horizontally extending vertically spaced panels and means to support them in the housing in spaced relation said panels each having a multiplicity of apertures therein arranged in the samepredetermined geometric patterns so that the corresponding apertures in the different panels are aligned, a multiplicity of tubes in the housing each tube extending through the corresponding apertures of the panels, weights slidable in geometric patterns as the apertures in said panels, said members extending through the apertures in said table section and having heads above the table section too large to pass through the apertures in said table section, and a support on said housing above the table top having means on which the heads of said members may be releasably mounted.

2. A device for demonstrating geometric figures comprising in combination, a housing, a framework therein comprising a plurality of horizontally extending vertically spaced panels and means to support them in the housing in spaced relation said panels each having a multiplicity of apertures therein arranged in the same predetermined geometric patterns so that the corresponding apertures in the difierent panels are aligned, a multiplicity of tubes in the housing each tube extending through the corresponding apertures of they panels, weights slidable in said tubes, flexible members secured to said weights, and projecting from the upper ends of said tubes, a table section over said housing having apertures therein over said tubes, the apertures being arranged in the same predetermined geometric patterns as the apertures in said panels, said members extending through the apertures in said table section and having heads above the table section too large to pass through the apertures in said table section, each of the tubes being located under an aperture, certain apertures in the table section having more than one tube under them with flexible members from all such tubes extending through the same aperture, and a support on said housing above the table top having means on which the heads of said members may be releasably mounted.

3. A device for demonstrating geometric figures comprising in combination, a housing, upright means thereon for supporting rods, and the like, above the housing, a removable table top on the housing, said top having a multiplicity of apertures therein arranged in a predetermined geometric pattern radially and circumferentially about a central point on said table, tubular string guides under the apertures, means supporting said tubular guides in upright position in the housing, strings extending through said apertures and having heads thereon over the table, and means in said tubular guides pulling the strings downward, the means supporting said guides comprising vertically spaced horizontal panels in said housing, said panels having aligned apertures therein through which the tubular guides are vertically movable, certain aligned apertures in said panels having more than one tubular guide therein.

FLOYD P. ARMSTEAD.

REFERENCES CITED UNITED STATES PATENTS 5 Number Name Date 630,217 Hanstein Aug. 1, 1899 2,168,634 Spencer Aug. 8, 1939 2,312,175 Korotzer Feb. 23, 1943 2,459,749 Bosomworth Jan. 18, 1949 

